Question: You have found the following ages (in years) of all 4 snakes at your local zoo: $ 12,\enspace 5,\enspace 5,\enspace 16$ What is the average age of the snakes at your zoo? What is the standard deviation? You may round your answers to the nearest tenth.
Solution: Because we have data for all 4 snakes at the zoo, we are able to calculate the population mean $({\mu})$ and population standard deviation $({\sigma})$ To find the population mean , add up the values of all $4$ ages and divide by $4$ $ {\mu} = \dfrac{\sum\limits_{i=1}^{{N}} x_i}{{N}} = \dfrac{\sum\limits_{i=1}^{{4}} x_i}{{4}} $ $ {\mu} = \dfrac{12 + 5 + 5 + 16}{{4}} = {9.5\text{ years old}} $ Find the squared deviations from the mean for each snake. Age $x_i$ Distance from the mean $(x_i - {\mu})$ $(x_i - {\mu})^2$ $12$ years $2.5$ years $6.25$ years $^2$ $5$ years $-4.5$ years $20.25$ years $^2$ $5$ years $-4.5$ years $20.25$ years $^2$ $16$ years $6.5$ years $42.25$ years $^2$ Because we used the population mean $({\mu})$ to compute the squared deviations from the mean , we can find the variance $({\sigma^2})$ , without introducing any bias, by simply averaging the squared deviations from the mean $ {\sigma^2} = \dfrac{\sum\limits_{i=1}^{{N}} (x_i - {\mu})^2}{{N}} $ $ {\sigma^2} = \dfrac{{6.25} + {20.25} + {20.25} + {42.25}} {{4}} $ $ {\sigma^2} = \dfrac{{89}}{{4}} = {22.25\text{ years}^2} $ As you might guess from the notation, the population standard deviation $({\sigma})$ is found by taking the square root of the population variance $({\sigma^2})$ ${\sigma} = \sqrt{{\sigma^2}}$ $ {\sigma} = \sqrt{{22.25\text{ years}^2}} = {4.7\text{ years}} $ The average snake at the zoo is 9.5 years old. There is a standard deviation of 4.7 years.